Cellular automaton model of self-healing

TL;DR

This paper introduces a cellular automaton model for self-healing systems, demonstrating its ability to repair localized damage with high efficiency, especially for small damage sizes, and analyzing its limitations for larger damages.

Contribution

The paper presents a simple, rule-based cellular automaton model that effectively repairs localized damage, providing insights into its efficiency and limitations for different damage sizes.

Findings

The model always repairs single-color damage.

High repair efficiency (>98%) for damage within 5x5 area.

Efficiency decreases with larger damage sizes, approaching zero as size increases.

Abstract

We propose a simple cellular automaton model of a self-healing system and investigate its properties. In the model, the substrate is a two-dimensional checkerboard configuration which can be damaged by changing values of a finite number of sites. The cellular automaton we consider is a checkerboard voting rule, a binary rule with Moore neighbourhood which is topologically conjugate to majority voting rule. For a single color damage (when only cells in the same state are modified), the rule always fixes the damage. For a general damage, when it is localized inside a $3 \times 3$ square, the rule also fixes it always. When the damage is inside of a larger $n \times n$ square, the efficiency of the rule in fixing the damage becomes smaller than $100\%$, but it remains better than $98\%$ for $n \leq 5$ and better than $75 \%$ for $n\leq 7$. We show that in the limit of infinite $n$ the efficiency tends to zero.